I think you should define the tensor Omega[-a, -b] and its inverse, say iOmega[a, b], as separate antisymmetric tensors, and use them systematically with those index configurations. Because there is no (Riemannian) metric, you cannot raise and lower their indices.

To help with the visualization, I'm making both Omega and iOmega format as \[CapitalOmega]:

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